I need the answers for 1.4-11 (c) Are the events \"A fails and\"B fasls statlsti
ID: 3071302 • Letter: I
Question
I need the answers for 1.4-11
(c) Are the events "A fails and"B fasls statlstieany muependent 1411. The communication system of Example 1.4-2 is to be extended to the case of three transmitted symbols 0, 1, and 2. Define appropriate events A, and B, i . 2, to represent symbols after and before the channel, respectively. Assume channe transition probabilities are all equal at P(AM)-ol, i tj. and are PC,B) a 0.8 for ,2, 3, while symbol transmission probabilities are PB)-0.5, P8,)0 and P(B, 0.2 (a) Sketch the diagram analogous to Fig. 1.4-2 (b) Compute received symbol probabilities PLA)P) and PA,). (c) Compute the a posteriori probabilities for this system. (d) Repeat parts (b) and (c) for all transmission symbol probabilities equal. Note the effect botale Tuo nroduction lines usedExplanation / Answer
P(B1) = 0.5
P(B2) = 0.3
P(B3) = 0.2
P(A1|B1) = 0.8
P(A2|B1) = 0.1
P(A3|B1) = 0.1
P(A1|B2) = 0.1
P(A2|B2) = 0.8
P(A3|B2) = 0.1
P(A1|B3) = 0.1
P(A2|B3) = 0.1
P(A3|B3) = 0.8
therefore, P(A1) = 0.45
P(A2) = 0.31
P(A3) = 0.24
c. P(Bi|Aj) = P(Bi and Aj) / P(Aj)
P(B1|A1) = 0.4 / 0.45
P(B2|A1) = 0.03 / 0.45
P(B3|A1) = 0.02 / 0.45
P(B1|A2) = 0.05 / 0.31
P(B2|A2) = 0.24 / 0.31
P(B3|A2) = 0.02 / 0.31
P(B1|A3) = 0.05 / 0.24
P(B2|A3) = 0.03 / 0.24
P(B3|A3) = 0.16 / 0.24
d.
P(Ai) = 1/3 for i = 1,2,3
P(Bj) = 1/3 for j = 1,2,3
P(A1|B1) = 0.8
P(A2|B1) = 0.1
P(A3|B1) = 0.1
P(A1|B2) = 0.1
P(A2|B2) = 0.8
P(A3|B2) = 0.1
P(A1|B3) = 0.1
P(A2|B3) = 0.1
P(A3|B3) = 0.8
Now.
P(Bi|Aj) = P(Bi and Aj) / P(Aj)
P(B1|A1) = 0.267 / 1/3
P(B2|A1) = 0.033 / 1/3
P(B3|A1) = 0.033 / 1/3
P(B1|A2) = 0.033 / 1/3
P(B2|A2) = 0.267 / 1/3
P(B3|A2) = 0.033 / 1/3
P(B1|A3) = 0.033 / 1/3
P(B2|A3) = 0.033 / 1/3
P(B3|A3) = 0.267 / 1/3
P(Ai and Bj) = P(Bj) * P(Ai|Bj) A 1 2 3 B 1 0.5 * 0.8 0.5 * 0.1 0.5 * 0.1 2 0.3 * 0.1 0.3 * 0.8 0.3 * 0.1 3 0.2 * 0.1 0.2 * 0.1 0.2 * 0.8